Before the Lapua was being produced in factory rifles the .338 edge seemed to be the big bully on the block.
Most of the Savages with this conversion that I read about were built using 300RUMs and the only thing necessary was a barrel change as it had the right mag cutout length and magazine to run the cartridge.
Of course most guys took the upgrades a lot further but the most economical way I know to do it would be to buy one someone has already built or find a 300RUM to start with and have a barrel built.
There are a few Savages in .300RUM that show up on the big auction site from time to time that would be great donors.
As for me I parked my lefty 116 in RUM long ago. Just waaay too much rifle for my purposes.
Good luck